Introduction to Probability with Texas Hold'em Examples

197 Pages·2011·1.28 MB·English

by  Schoenberg

I
N
Statistics T
R
O
I N T R O D U C T I O N T O D I N T R O D U C T I O N T O
U
C
P R O B A B I L I T Y W I T H T P R O B A B I L I T Y W I T H
I
O
N
T E X A S H O L D ’ E M
T T E X A S H O L D ’ E M
O

P
E X A M P L E S R
O E X A M P L E S
B
A
B
Introduction to Probability with Texas Hold’em Examples illustrates I
L
both standard and advanced probability topics using the popular I
T
Y
poker game of Texas Hold’em, rather than the typical balls in urns.

W
The author uses students’ natural interest in poker to teach important
I
T
concepts in probability. H

T
This classroom-tested book covers the main subjects of a standard E
X
undergraduate probability course, including basic probability rules,
A
standard models for describing collections of data, and the laws of S

H
large numbers. It also discusses several more advanced topics, such O
L
as the ballot theorem, the arcsine law, and random walks, as well as
D
some specialized poker issues, such as the quantification of luck and ’
E
M
skill in Texas Hold’em. Homework problems are provided at the end

E
of each chapter.
X
A
The author includes examples of actual hands of Texas Hold’em M
P
from the World Series of Poker and other major tournaments and L
E
televised games. He also explains how to use R to simulate Texas
S
Hold’em tournaments for student projects. R functions for running the
S
tournaments are freely available from CRAN.
C
H F R E D E R I C PA I K S C H O E N B E R G
O
E
N
B
E
K11367 R
G
K11367_Cover.indd 1 11/3/11 10:55 AM
I N T R O D U C T I O N T O
P R O B A B I L I T Y W I T H
T E X A S H O L D ’ E M
E X A M P L E S

I N T R O D U C T I O N T O
P R O B A B I L I T Y W I T H
T E X A S H O L D ’ E M
E X A M P L E S
FREDERIC PAIK SCHOENBERG
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2011 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20150121
International Standard Book Number-13: 978-1-4398-2769-7 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-
ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface vii
Chapter 1 Probability Basics 1
1.1 Meaning of Probability 1
1.2 Basic Terminology 2
1.3 Axioms of Probability 3
1.4 Venn Diagrams 4
1.5 General Addition Rule 5
Exercises 8
Chapter 2 Counting Problems 11
2.1 Sample Spaces with Equally Probable Events 11
2.2 Multiplicative Counting Rule 15
2.3 Permutations 16
2.4 Combinations 19
Exercises 35
Chapter 3 Conditional Probability and Independence 39
3.1 Conditional Probability 39
3.2 Independence 44
3.3 Multiplication Rules 45
3.4 Bayes’s Rule and Structured Hand Analysis 49
Exercises 53
Chapter 4 Expected Value and Variance 57
4.1 Cumulative Distribution Function and
Probability Mass Function 57
4.2 Expected Value 58
4.3 Pot Odds 64
4.4 Luck and Skill in Texas Hold’em 71
4.5 Variance and Standard Deviation 79
4.6 Markov and Chebyshev Inequalities 82
4.7 Moment-Generating Functions 84
Exercises 85
v
vi   ■   Contents
Chapter 5 Discrete Random Variables 91
5.1 Bernoulli Random Variables 91
5.2 Binomial Random Variables 93
5.3 Geometric Random Variables 95
5.4 Negative Binomial Random Variables 97
5.5 Poisson Random Variables 98
Exercises 101
Chapter 6 Continuous Random Variables 103
6.1 Probability Density Functions 103
6.2 Expected Value, Variance, and Standard
Deviation 105
6.3 Uniform Random Variables 107
6.4 Exponential Random Variables 114
6.5 Normal Random Variables 115
6.6 Pareto Random Variables 117
6.7 Continuous Prior and Posterior Distributions 120
Exercises 122
Chapter 7 Collections of Random Variables 125
7.1 Expected Values and Variances of Sums
of Random Variables 125
7.2 Conditional Expectation 128
7.3 Laws of Large Numbers and the Fundamental
Theorem of Poker 130
7.4 Central Limit Theorem 136
7.5 Confidence Intervals for the Sample Mean 140
7.6 Random Walks 145
Exercises 154
Chapter 8 Simulation and Approximation Using Computers 157
Exercises 164
Appendix A: Abbreviated Rules of Texas Hold’em 167
Appendix B: Glossary of Poker Terms 171
Appendix C: Solutions to Selected Odd-Numbered Exercises 175
References and Suggested Reading 183
Index 185
Preface
I am a lousy poker player. Let me get that out of the way right
off the bat. If you are reading this book in the hope that you will
learn strategy tips on how to be a better poker player, you are
bound to be disappointed. This is not a book on how to use proba-
bility to play Texas Hold’em. It is a textbook using Texas Hold’em
examples to teach probability.
The other thing I want to state right from the outset is that I
in no way intend this book to be an endorsement of gambling.
Poker, like all forms of gambling, can be addictive and dangerous.
The morality of gambling has been properly questioned by many
for a host of reasons, and among them is the fact that many peo-
ple, especially those who can least afford it, often lose more than
they should prudently risk. The rise of online gambling recently,
especially among students at colleges and universities, is cause
for serious concern. When I have taught my course on poker and
probability at UCLA in the past, I have always started out on the
first day by lecturing about the dangers of gambling, and my first
required readings for the students are handouts on the perils of
gambling addiction.
The purpose of this book is not to promote gambling or to
teach how to play poker. Instead, my intention is to use students’
natural interest in poker to motivate them to learn important top-
ics in probability. The first few times I taught probability, I was
disappointed by many of the examples in the books. They typi-
cally involved socks in sock drawers or balls in urns. Most of my
students did not even know what an urn was, and certainly were
not motivated when informed of its meaning. I thought it would
be interesting to try to teach the same topics as those covered
in most probability texts using only examples from poker. I was
happy to find that students seemed to vastly prefer these examples
and that it was hardly a challenge at all to motivate even quite
complex subjects using poker. In fact, I needed to look no fur-
ther than Texas Hold’em, which is currently the most popular
poker game, to illustrate all the standard undergraduate probabil-
ity topics and even some more advanced topics. While some have
urged me to discuss other poker games, I have decided instead to
stick exclusively to Texas Hold’em examples, for two reasons. The
first is that Texas Hold’em is more popular and more commonly
vii
viii   ■   Preface
televised than other games and hence may be of greater interest to
students. The second reason is simply brevity. The purpose of this
book is to teach probability, not the rules and intricacies of various
poker games, and I found absolutely no reason to search beyond
Texas Hold’em to illustrate any probability topic.
The topics covered in this book are similar to those in most
undergraduate probability textbooks, with a few exceptions. I
have added sections on special topics, including a few specialized
poker issues such as the quantification of luck and skill in Texas
Hold’em, and some topics typically found in graduate probability
texts, such as the ballot theorem and the arcsine law.
I will doubtless be criticized by some of my colleagues for writ-
ing this book, because poker is not only perceived as immoral but
also as frivolous. While many probabilists and statisticians might
feel that probability should be taught using more serious, scien-
tific examples, I disagree. I fully acknowledge the downsides of
poker, but poker has its good qualities as well. Texas Hold’em is
fun, and its current popularity can be used to attract students and
keep them interested. Texas Hold’em involves a blend of luck and
skill that may be extremely frustrating at times for players, but
can also be wonderfully intriguing, and incidentally is in many
ways similar to other pursuits in life that seem to rely on a similar
blend of skill and fortune, such as a search for a job or for love.
Most importantly, in my opinion, Texas Hold’em is at its heart an
intellectual pursuit. Gambling games such as poker have inspired
many of the most important ideas in probability theory, including
Bayes’s theorem and the laws of large numbers that have found
applications in so many scientific disciplines.
Aside from using exclusively Texas Hold’em examples, one
other feature that I hope may make this book unique as a prob-
ability textbook is that I have tried, wherever possible, to use only
real examples—not realistic, but real examples from actual hands
of Texas Hold’em shown played in the World Series of Poker or
other major tournaments or televised games. The search for these
hands was time-consuming but enjoyable, and the use of real
examples may help to keep students interested. Sometimes the
probability topic discussed is somewhat tangent to the main issue
that makes a poker hand interesting, but hopefully readers can
look past this.
When I have taught this course in the past, in addition to
homework and exams, I assigned the students two computational
Preface   ■   ix
Nine-month-old poker players Max (left), and Gemma (right) Paik Schoenberg,
May 2010.
projects. On the first project, the students were asked to write a
function in R that took various inputs including their cards, the
betting before them, their number of chips, the number of players
at the table, and the size of the blinds, and then output a bet size
of 0 or their number of chips. That is, they had to write a program
to fold or go all-in. I would then have their computer programs
compete in tournaments that I ran multiple times. For the final
project, they were asked to write a Texas Hold’em program in
R that could be more complicated, and did not require them to
go all-in or fold, but allowed them to bet intermediate amounts.
Some students sincerely enjoyed these projects and wrote quite
elaborate functions. Several students felt that this was their favor-
ite aspect of the course. I have compiled functions for instructors
to use to run these tournaments, as well as some examples of the
students’ functions, into a public R package called holdem which
may be freely downloaded from www.r-project.org, and some
description is given in Chapter 8 as well.
I have a lot of people to thank. First and foremost, I thank
my wife, Jean, not only for supporting me throughout the
writing of this book but also for indirectly introducing me to
Texas Hold’em by taking me on a surprise trip to Las Vegas
for my birthday several years ago. Gamma, Dad, Mom, Randy,